We investigate the algebra and geometry of general interventions in discrete DAG models. To this end, we introduce a theory for modeling soft interventions in the more general family of staged tree models and develop the formalism to study these models as parametrized subvarieties of a product of probability simplices. We then consider the problem of finding their defining equations, and we derive a combinatorial criterion for identifying interventional staged tree models for which the defining ideal is toric. We apply these results to the class of discrete interventional DAG models and establish a criteria to determine when these models are toric varieties.

翻译：我们研究离散有向无环图模型中一般干预的代数与几何。为此，我们引入了一种在更广泛的阶段树模型族中建模软干预的理论，并建立了将这些模型视为概率单纯形乘积中参数化子流形的形式化方法。进而考虑寻求其定义方程的问题，并推导出判定干预阶段树模型定义理想为环面簇的组合学准则。我们将这些结果应用于离散干预有向无环图模型类，建立了判定这些模型何时为环面簇的准则。